Sparse analytic systems
Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal {F}$ of (real or complex) analytic functions, such that $\big \{ f(x) \ : \ f \in \mathcal {F} \big \}$ is countable for every x. We strengthen Erdős’ result by proving...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423000543/type/journal_article |
| _version_ | 1797787964928950272 |
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| author | Brent Cody Sean Cox Kayla Lee |
| author_facet | Brent Cody Sean Cox Kayla Lee |
| author_sort | Brent Cody |
| collection | DOAJ |
| description | Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family
$\mathcal {F}$
of (real or complex) analytic functions, such that
$\big \{ f(x) \ : \ f \in \mathcal {F} \big \}$
is countable for every x. We strengthen Erdős’ result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. We use such systems to construct, assuming CH, an equivalence relation
$\sim $
on
$\mathbb {R}$
such that any ‘analytic-anonymous’ attempt to predict the map
$x \mapsto [x]_\sim $
must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman [2]. |
| first_indexed | 2024-03-13T01:29:27Z |
| format | Article |
| id | doaj.art-cb910b5dac9c4b1fa74441134731e761 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-13T01:29:27Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-cb910b5dac9c4b1fa74441134731e7612023-07-04T09:18:29ZengCambridge University PressForum of Mathematics, Sigma2050-50942023-01-011110.1017/fms.2023.54Sparse analytic systemsBrent Cody0https://orcid.org/0000-0001-5890-5222Sean Cox1https://orcid.org/0000-0001-5546-7079Kayla Lee2Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, VA 23284, USA; E-mail:Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, VA 23284, USA; E-mail:Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, VA 23284, USA; E-mail:Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal {F}$ of (real or complex) analytic functions, such that $\big \{ f(x) \ : \ f \in \mathcal {F} \big \}$ is countable for every x. We strengthen Erdős’ result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. We use such systems to construct, assuming CH, an equivalence relation $\sim $ on $\mathbb {R}$ such that any ‘analytic-anonymous’ attempt to predict the map $x \mapsto [x]_\sim $ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman [2].https://www.cambridge.org/core/product/identifier/S2050509423000543/type/journal_article03E5003E2526E0530D20 |
| spellingShingle | Brent Cody Sean Cox Kayla Lee Sparse analytic systems Forum of Mathematics, Sigma 03E50 03E25 26E05 30D20 |
| title | Sparse analytic systems |
| title_full | Sparse analytic systems |
| title_fullStr | Sparse analytic systems |
| title_full_unstemmed | Sparse analytic systems |
| title_short | Sparse analytic systems |
| title_sort | sparse analytic systems |
| topic | 03E50 03E25 26E05 30D20 |
| url | https://www.cambridge.org/core/product/identifier/S2050509423000543/type/journal_article |
| work_keys_str_mv | AT brentcody sparseanalyticsystems AT seancox sparseanalyticsystems AT kaylalee sparseanalyticsystems |